The Busemann theorem for complex p-convex bodies

p-convex bodies

HUANG Qingzhong, HE Binwu and WANG Guangting

Abstract. The Busemann theorem states that the intersection body of an origin-symmetric convex body is also convex. In this paper, we prove a version of Busemann theorem for complexp-convex bodies. Namely, the complex intersection body of an origin-symmetric complexp-convex body is γ-convex for certain γ. The result is the complex analogue of Kim, Yaskin, Zvavitch’s work on (real)p-convex bodies. Furthermore, we show that the generalized radialqth mean body of ap-convex body isγ-convex for certainγ.

Mathematics Subject Classification (2010).52A20.

Keywords.p-convex body, intersection body, complex intersection body, generalized radialqth mean body, Busemann theorem.

1. Introduction

As usual, Sⁿ⁻¹ denotes the unit sphere and o the origin in Euclidean n- spaceRⁿ. Ifξ∈Sⁿ⁻¹, we denote byξ^⊥the central hyperplane perpendicular to the vectorξ. We use the notation|K|for the volume of a compact setK.

A body is a compact set with nonempty interior. A setKis star-shaped with respect to the origin if every line through the origin which meetsKdoes so in a closed line segment. LetK⊂Rⁿ be a star shaped body (with respect to the origin), the radial functionρ(K,·) :Rⁿ\{o} →Ris defined by

ρ(K, x) =ρK(x) = max{λ≥0 :λx∈K}.

The Minkowski functionalk · kK :Rⁿ→Ris defined by kxkK = min{λ≥0 : x∈λK}. Obviously,ρK(x) =kxk⁻¹_K , forx∈Rⁿ\{o}.

The concept of an intersection body was introduced by Lutwak [15] in 1988. LetKandLbe origin symmetric star bodies inRⁿ. We say thatKis the intersection body ofLand writeK=I(L) if the radius ofKin every direction

The authors would like to acknowledge the support from the National Natural Science Foundation of China (11071156), Shanghai Leading Academic Discipline Project (J50101).

is equal to the volume of the central hyperplane section of L perpendicular to this direction, i.e. for everyξ∈Sⁿ⁻¹,

kξk⁻¹_I(L)=|L∩ξ^⊥|.

For additional references regarding intersection bodies and their applications, the reader may wish to consult the books by Gardner [12] and Koldobsky [5].

One celebrated result for intersection bodies is the classical Busemann theorem (see [3, 12, 16]):

Theorem 1.1. Let K be an origin-symmetric convex body in Rⁿ. Then the intersection bodyI(K) ofK is an origin symmetric convex body.

Letp∈(0,1]. A bodyK isp-convex if, for all x, y∈Rⁿ,

kx+yk^p_K≤ kxk^p_K+kyk^p_K, (1.1) or, equivalentlyt^1/px+ (1−t)^1/py∈Kfor any pointsx, y∈Kandt∈(0,1).

Note that p-convex sets with p= 1 are just convex. Moreover, a p1-convex body isp2-convex for all 0< p2≤p1.

Recently, a version of the Busemann theorem forp-convex bodies was obtained by Kim, Yaskin, Zvavitch [4], which can be formulated as follows:

Theorem 1.2. Let K be an origin-symmetric p-convex body in Rⁿ for p ∈ (0,1]. Then the intersection bodyI(K)ofKis an origin symmetricγ-convex body withγ= ((1/p−1)(n−1) + 1)⁻¹.

The study of complex convex bodies just appears recent years, and results appear only occasionally (see, e.g., [1, 6, 7, 8, 9, 10, 14, 17, 18]). To formulate the complex version, we need several definitions.

Origin-symmetric convex bodies inCⁿare the unit balls of norms onCⁿ. We denote byk · kK the norm corresponding to the bodyK:

K={z∈Cⁿ:k · kK ≤1}.

We identifyCⁿ withR²ⁿ using the standard mapping

ξ= (ξ1,· · ·ξn) = (ξ11+iξ12,· · ·, ξn1+iξn2)7→(ξ11, ξ12,· · ·, ξn1, ξn2).

Since norms onCⁿ satisfy the equality

kλzk=|λ|kzk, ∀z∈Cⁿ,∀λ∈C,

origin symmetric complex convex bodies correspond to those origin symmet- ric convex bodiesKinR²ⁿthat are invariant with respect to any coordinate- wise two-dimensional rotation, namely for eachθ∈[0,2π] and each (ξ11, ξ12,

· · ·ξn1, ξn2)∈R²ⁿ

kξkK=kRθ(ξ11, ξ12),· · ·, Rθ(ξn1, ξn2)kK, (1.2) whereRθ stands for the counterclockwise rotation ofR² by the angleθwith respect to the origin. We shall say thatKis a complex convex body inR²ⁿifK is a convex body and satisfies equations (1.2).

Forξ∈Cⁿ, |ξ|= 1, denote by H_ξ ={z∈Cⁿ: (z, ξ) =

n

X

k=1

z_kξ_k = 0}

the complex hyperplane through the origin, perpendicular to ξ. Under the standard mapping fromCⁿ toR²ⁿ the hyperplaneH_ξ turns into a (2n−2)- dimensional subspace ofR²ⁿ orthogonal to the vectors

ξ= (ξ11, ξ12,· · ·, ξn1, ξn2) and ξ^⊥ = (−ξ12, ξ11,· · · ,−ξn2, ξn1).

The orthogonal two-dimensional subspaceH_ξ^⊥ has orthonormal basis ξ, ξ^⊥. A star (convex) bodyK inR²ⁿ is a complex star (convex) body if and only if, for everyξ∈S²ⁿ⁻¹, the sectionK∩H_ξ^⊥ is a two-dimensional Euclidean circle with radiusρ_K(ξ) =kξk⁻¹_K .

Corresponding to the (real) intersection body, the complex counterpart was introduced recently by Kolodobsky, Paouris and Zymonopoulou [9].

Definition 1.3. LetK, Lbe origin-symmetric complex star bodies inR²ⁿ. We say thatK is the complex intersection body ofL and writeK=Ic(L) if for everyξ∈R²ⁿ

|K∩H_ξ^⊥|=|L∩Hξ|. (1.3) Since K∩H_ξ^⊥ is the two-dimensional Euclidean circle with radius kξk⁻¹_K , (1.3) can be written as

πkξk⁻²_I

c(L)=|L∩Hξ|. (1.4)

In [9], Kolodobsky, Paouris and Zymonopoulou also proved the complex version of the Busemann theorem.

Theorem 1.4. Let Kbe an origin-symmetric complex convex body in Cⁿ and Ic(K)the complex intersection body ofK. ThenIc(K)is also an origin sym- metric convex body inCⁿ.

The main purpose of this paper is to give a version of the Busemann theorem for complexp-convex bodies, corresponding to the result for (real)p- convex bodies.

Theorem 1.5. LetKbe an origin-symmetric complexp-convex body inCⁿ and Ic(K)the complex intersection body ofK. ThenIc(K)is also an origin sym- metricγ−convex body in Cⁿ with γ= ((1/p−1)(n−1) + 1)⁻¹.

Clearly, Theorem 1.5 reduces to Theorem 1.4 if p= 1. From Theorem 1.2 and Theorem 1.5, we see that given ap-convex body, the corresponding intersection body in real and complex spaces are both γ-convex with γ = ((1/p−1)(n−1) + 1)⁻¹. In [4], Kim, Yaskin, Zvavitch obtained that thisγis asymptotically optimal in real space. However, the problems of the exact value ofγ in real and complex spaces remain open.

The rest of this paper is organized as follows: In Section 2, a basic in- equality is established, which is crucial for our arguments. Section 3 contains the proof of the Busemann theorem for (real) p-convex bodies due to Kim,

Yaskin, Zvavitch [4]. In Section 4, analogously to the real case, we establish the Busemann theorem for complexp-convex bodies. Moreover, in Section 5 we introduce the concept of a generalized radialqth mean body, and prove that the generalized radialqth mean body of ap-convex body isγ-convex for certainγ, extending a result of Gardner and Zhang [13].

2. A Basic inequality

The following result given by Kolodobsky, Paouris and Zymonopoulou [9] is a variant origining from Busemann [3] and Ball [2].

Theorem 2.1. Let r1, r2>0 and leta >0. Definet1, t2, r3 by:

t₁:= r₂ r1+r2

, t₂:= r₁ r1+r2

, (2.1)

a r₃ = 1

r₁ + 1

r₂. (2.2)

Supposef1, f2, f3: [0,∞)→[0,∞)are three integrable functions, such that f3(r3)≥f₁^t1(r1)f₂^t2(r2), ∀r1, r2≥0. (2.3) Let q >0 and denote

F₁:=Z ∞ 0

r^q−1f₁(r)dr¹_q , F₂:=Z ∞

0

r^q−1f₂(r)dr¹_q , F₃:=Z ∞

0

r^q−1f₃(r)dr¹_q . Then,

a F3

≤ 1 F1

+ 1 F2

. (2.4)

Inspired by the results of Kim, Yaskin, Zvavitch [4], we extend the above inequality to the inequality forp-convex bodies, which can be stated as fol- lows:

Theorem 2.2. Let r1, r2>0 and leta, p >0. Definet1, t2, r3 by:

t1:= r^−p₁

r^−p₁ +r^−p₂ , t2:= r₂^−p

r₁^−p+r₂^−p, (2.5) a^pr₃^−p=r^−p₁ +r₂^−p. (2.6) Supposef1, f2, f3 : [0,∞)→[0,∞) are three integrable functions, and there existsα≥0 such that

f3(r3)≥(t^t₁¹t^t₂²)^αf₁^t1(r1)f₂^t2(r2), ∀ r1, r2≥0. (2.7)

Let q >0 and denote

F₁:=Z ∞ 0

r^q−1f₁(r)dr¹_q , F2:=Z ∞

0

r^q−1f2(r)dr¹_q , F3:=Z ∞

0

r^q−1f3(r)dr¹_q . Then,

a^γ F₃^γ ≤ 1

F₁^γ + 1

F₂^γ, (2.8)

whereγ= (α/q+ 1/p)⁻¹.

Note that Theorem 2.2 reduces to Theorem 2.1 by settingp= 1, α= 0.

Proof. For everys∈[0,1] we defineri=ri(s) fori= 1,2 by s= 1

F₁^q Z r₁

0

r^q−1f1(r)dr= 1 F₂^q

Z r₂

0

r^q−1f2(r)dr, then fori= 1,2

dri

ds = F_i^q r_i^q−1fi(ri). Denoter₃=r₃(s) by (2.6), then

dr3

ds =a^−pr₃^p+1

r₁^−p−1dr1

ds +r^−p−1₂ dr2

ds

=r3

h t1(1

r1

dr1

ds) +t2(1 r2

dr2

ds)i

≥r3

1 r1

dr1

ds ^t11

r2

dr2

ds ^t2

=a t

1 p

1

dr1

ds ^t1

t

1 p

2

dr2

ds ^t2

=a(t^t₁¹t^t₂²)^1p F₁^q r^q−1₁ f₁(r₁)

t1 F₂^q r^q−1₂ f₂(r₂)

t2

. Together with (2.6) and (2.7), it follows that

Z ∞

0

r^q−1f3(r)dr≥ Z 1

0

r₃^q−1f3(r3)dr3

dsds

≥a^q Z 1

0

(t^t₁¹t^t₂²)^α+pqF₁^qt1F₂^qt2ds

=a^q Z 1

0

h

(t1F₁^qm)^t1(t2F₂^qm)^t2i_m¹ ds

≥a^q Z 1

0

h t₁

t1F₁^qm + t₂ t2F₂^qm

i−_m¹

ds

=a^qh 1 F₁^qm + 1

F₂^qm i−_m¹

,

where m = (α+q/p)⁻¹. Therefore, (2.8) follows with γ = qm = (α/q+

1/p)⁻¹.

The following Lemma is crucial to connect Theorem 2.2 and the γ- convexity of a body, which was first observed by Kim, Yaskin, Zvavitch [4].

We present their result with minor modifications.

Lemma 2.3. LetK be ap-convex body and letLbe ak-dimensional p-convex set in Rⁿ, and the function f(x) be given by f(x) = |K∩(L+x)|. Then for x1, x2 ∈ Rⁿ, x3 = a⁻¹(x1+x2) and t1, t2, r3 satisfying (2.5), (2.6), we have that

f(r₃x₃)≥(t^t₁¹t^t₂²)^k(1p−1)f(r₁x₁)^t1f(r₂x₂)^t2. (2.9) Proof. Note thatr₃x₃=t

1 p

1r₁x₁+t

1 p

2r₂x₂. Then K∩(L+r3x3) =K∩(L+t

1 p

1r1x1+t

1 p

2r2x2)

⊃t

1 p

1

K∩(L+r1x1) +t

1 p

2

K∩(L+r2x2)

=t1

t

1 p−1

1 (K∩(L+r1x1)) +t2

t

1 p−1

2 (K∩(L+r2x2)) . Using the Brunn-Minkowski inequality (see [11]), we have

f(r3x3)≥ t

1 p−1

1 (K∩(L+r1x1))

t₁ t

1 p−1

2 (K∩(L+r2x2))

t₂

= (t^t₁¹t^t₂²)^k(1p−1)

K∩(L+r₁x₁)

t₁

K∩(L+r₂x₂)

t₂

= (t^t₁¹t^t₂²)^k(1p−1)f(r1x1)^t1f(r2x2)^t2.

3. The Busemann theorem for p-convex bodies

The following results were first obtained by Kim, Yaskin, Zvavitch [4] along the same approach. For reader’s convenience, we present it here.

Lemma 3.1. Let K be an origin-symmetric p-convex body in Rⁿ, p∈(0,1], andE an (n−2)-dimensional subspace ofRⁿ. Let u∈E^⊥ be a unit vector andr(u) :=|K∩span{u, E}|. Thenr:E^⊥∩Sⁿ⁻¹→(0,∞)is the boundary of aγ-convex body inH^⊥ with γ= ((1/p−1)(n−1) + 1)⁻¹.

Proof. Note thatEis (n−2)-dimensionalp-convex set inRⁿ in Lemma 2.3.

Letu₁, u₂∈Sⁿ⁻¹∩E^⊥, and u₃:= _|u^u1+u2

1+u2| =:a⁻¹(u₁+u₂). Denote f_i(r) = 2|K∩(E+rui)|fori= 1,2,3. By (1.1), it is sufficient to show that

a^γ

r(u₃)^γ ≤ 1

r(u₁)^γ + 1

r(u₂)^γ. (3.1)

Observe that

r(u_i) = 2 Z ∞

0

|K∩(E+ru_i)|dr= Z ∞

0

f_i(r)dr.

Now, Lemma 2.3 withxi =uifori= 1,2,3 and Theorem 2.2 withq= 1 yield

(3.1).

Proof of Theorem 1.2.LetE be any (n−2)-dimensional subspace. Letu∈ Sⁿ⁻¹∩E^⊥, and letv∈Sⁿ⁻¹∩E^⊥ be orthogonal tou. Then

ρ_I(K)(v) =|K∩span{u, E}|.

From Lemma 3.1, we see thatI(K)∩E^⊥isγ-convex withγ= ((1/p−1)(n−

1) + 1)⁻¹. This impliesI(K) isγ-convex.

4. The Busemann theorem for complex p-convex bodies

In [9], Kolodobsky, Paouris and Zymonopoulou established the Busemann theorem for complex convex bodies. In the following, we will give the proof of the Busemann theorem for complexp-convex bodies (i.e. Theorem 1.5).

Recall that we identifyCⁿ withR²ⁿ. Assume thatn≥3. Letu1, u2∈ Cⁿ, |u1| = |u2| = 1, with H_u^⊥_i = span{ui, u^⊥_i }, θi ∈ S_H⊥

ui, i = 1,2. We defineu₃:= _|u^u1+u2

1+u2| ∈S_H⊥ u3

, withH_u^⊥

3 andθ₃:= _|θ^θ1+θ2

1+θ2| such that|θ₁+θ₂|=

|u1+u2|. We can assume thatH_u^⊥₁T

H_u^⊥₂ ={0}.

Letr1, r2>0. We definer3, ti such that fori= 1,2, ti:= r^−p_i

r₁^−p+r^−p₂ , r^p₃= |u1+u2|^p r^−p₁ +r₂^−p.

IfS := (H_u^⊥₁+H_u^⊥₂)^⊥, we denote Ei := span{H_u^⊥_i, S}, i= 1,2,3. The func- tiongi:H_u^⊥_i→R,fi: [0,∞)→[0,∞),i= 1,2,3 is defined by

gi(x) :=

Z

S+x

1K(y)dy=|K∩(S+x)|, f_i(r) :=g_i(rθ_i).

To prove Theorem 1.5, the following Lemma will be needed.

Lemma 4.1. ([9])Fori= 1,2,3, we have that

|K∩E_i|= 2π Z ∞

0

rf_i(r)dr.

Lemma 4.2. With the above notation, ifK is ap-convex body, then f3(r3)≥(t^t₁¹t^t₂²)^αf₁^t1(r1)f₂^t2(r1),

whereα= (1/p−1)(2n−4).

Proof. Note thatS is (2n−4)-dimensionalp-convex set inR²ⁿ. Takinga=

|θ1+θ2|=|u1+u2|and xi =θi for i= 1,2,3 in Lemma 2.3, we complete

the proof.

Theorem 4.3. With the above notation, if a:=|θ₁+θ₂|, then a^γ

|K∩E3|^γ/2 ≤ 1

|K∩E1|^γ/2+ 1

|K∩E2|^γ/2, whereγ= ((1/p−1)(n−1) + 1)⁻¹.

Proof. From Lemma 4.1, it follows that F_i²=

Z ∞

0

rfi(r)dr= |K∩Ei| 2π .

Then, applying Lemma 4.2 and Theorem 2.2 withq= 2, the result follows.

Corollary 4.4. Let K be an origin-symmetric complex p-convex body in Cⁿ. Let H be an (n−2)-dimensional subspace ofCⁿ. Let u∈H^⊥ be a complex unit vector and letHu:= span{H, u}andr(u) :=|K∩Hu|¹². Thenr:H^⊥∩ S²ⁿ⁻¹→(0,∞)is the boundary of a complexγ-convex body inH^⊥ withγ= ((1/p−1)(n−1) + 1)⁻¹.

Proof. Letu1, u2are two non-parallel unit vectors inH^⊥, andu3:= _|u^u1+u2

1+u₂|=:

a⁻¹(u₁+u₂). As in the proofs of Lemma 3.1, it is sufficient to show that a^γ

r(u3)^γ ≤ 1

r(u1)^γ + 1 r(u2)^γ.

In the notation of this section we have that H_ui =E_i, r(u_i) = |K∩E_i|¹².

The result follows from Theorem 4.3.

We are now ready to prove Theorem 1.5.

Proof of Theorem 1.5. In the case where n = 2 the body Ic(K) is simply a rotation of K, so the result is obvious. Let n ≥ 3. Then equation (1.4) and Corollary 4.4 imply that Ic(K)∩H^⊥ is γ-convex for every (n−2)- dimensional subspaceH ofCⁿ. This implies thatI_c(K) isγ-convex. Finally, it is not difficult to see thatI_c(K) satisfies (1.2). This implies thatI_c(K) is

a complexγ-convex body.

5. The generalized radial qth mean bodies R

(K )

LetKbe a body inRⁿ. The covariogramgK ofKis the function g_K(x) =|K∩(K+x)|, x∈Rⁿ.

The concept of a radialqth mean bodyRq(K) was introduced by Gar- nder and Zhang [13]. We introduce a generalization ofRq(K) for a bodyK, which is identical with the case that K is a convex body (see [13, Lemma 3.1]).

Definition 5.1. LetK be a body in Rⁿ and let q >0. Then the generalized radialqth mean bodyRq(K) is defined as the body whose radial function is given by

ρ_Rq(K)(x) = q

|K|

Z ∞

0

r^q−1g_K(rx)dr^1/q

, x∈Rⁿ\{o}.

Note that the above definition ofRq(K) is well defined, sinceρR_q(K)(x) is homogeneous of degree−1.

Lemma 5.2. Letf be a nonnegative integrable function inRⁿsatisfying (2.9).

Then the functionρdefined by ρ(x) =Z ∞

0

r^q−1f(rx)dr¹_q ,

for x ∈ Rⁿ\{o}, is the radial function of a γ-convex body in Rⁿ with γ = k

q(¹_p−1) +¹_p−1

.

Proof. Obviously,ρ(x) is homogeneous of degree−1. Forx1, x2∈Rⁿ, letx3= a⁻¹(x1+x2) and fi(r) = f(rxi) for i = 1,2,3. By Theorem 2.2 with α = k(¹_p−1), we obtain

a^γ

ρ^γ(x₃) ≤ 1

ρ^γ(x₁)+ 1 ρ^γ(x₂), whereγ=

k

q(¹_p−1) + ¹_p−1

. Consequently, 1

ρ^γ(x1+x2)≤ 1

ρ^γ(x1)+ 1 ρ^γ(x2).

The result follows from (1.1).

Theorem 5.3. If q >0, then the generalized radialqth mean bodyRq(K) of ap-convex bodyK,p∈(0,1], is an origin-symmetricγ-convex body withγ= n

q(¹_p −1) +¹_p−1

.

Proof. From Lemma 2.3, we see the covariogram g_K of K satisfies (2.9) with k=n. From Lemma 5.2 with f(rx) =qgK(rx)/|K|, we complete the

proof.

Corollary 5.4. ([13]) If q ≥ 0, then the radial qth mean body Rq(K) of a convex bodyK is an origin-symmetric convex body.

Acknowledgment

The authors wish to thank Professor A. Koldobsky for many valuable sug- gestions. We would also like to thank the referee for many helpful comments.

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HUANG Qingzhong Department of Mathematics Shanghai University Shanghai, 200444 China

e-mail:hqz376560571@163.com

HE Binwu

Department of Mathematics Shanghai University Shanghai, 200444 China

e-mail:hebinwu@shu.edu.cn

WANG Guangting

Department of Mathematics Shanghai University Shanghai, 200444 China

e-mail:guangtingw@gmail.com